Symplectic Analytical Singular Element For Plane Dynamic Fracture Analysis Via A Precise Algorithm In Time Domain

Fracture dynamics(dynamic fracture mechanics)is one of the branches of fracture mechanics.It is deeply related to many kinds of natural phenomena and engineering practices,and has vital importance both in theoretical and practical aspects.Now many analytical solutions are available for fracture dynamics problems with simple geometric and loading conditions.However,it is difficult to obtain analytical solutions in engineering practices,because of the complex geometric and loading conditions,as well as the interaction of stress waves.Therefore,numerical analysis has emerged an indispensable tool for fracture dynamics problems.In the point of view of numerical simulation,finite element method(FEM)is one of the most widely used methods and has many open sources and commercial software.Nevertheless,it is hard to obtain satisfactory results,when conventional finite elements are used to predict local stress singularity.Although a number of enhanced crack-tip elements have been developed,there are still some shortcomings,such as the need of transition elements,low efficiency and accuracy.Hence,it is of vital importance to further develop an effective crack-tip element for fracture dynamics problems.In this doctoral dissertation,the symplectic analytical singular element(SASE)originally used for the analysis of static crack problem,is combined with a precise algorithm in time domain for dynamic crack initiation problems.The main contents of this thesis includes:(1)Research background and significance of this thesis are introduced,as well as some important concepts.Furthermore,Research Status of analytical method and numerical method for dynamic fracture problem at home and abroad are presented in detail.(2)The symplectic system for plane static problem in polar coordinates is reviewed.Eigen solutions of sectorial domain problem are given,as well as eigen solutions and special solutions of bi-material interface crack problem.Additionally,special solution for V-notch subjected to arbitrary distributed load on cut surfaces is derived in symlpectic system.Specifically,the load is approximated by polynomial series,and special solution of each expanding term is specified analytically.Then the final special solution can be obtained accordingly.(3)Plane dynamic problem of structure with crack and V-notch is discussed.Firstly,the precise algorithm in time domain is introduced.Within each time interval,all time dependent physical quantities are expanded into series.Then governing equations and boundary conditions are transformed into recursive forms,expressed by expansion coefficients of the physical quantities.Consequently,the recursive FEM-based formulation is derived through weighted residual method.Secondly,SASE is constructed based on symplectic eigen solutions and special solutions for plane static fracture problem.Element stiffness matrix,Element mass matrix and Element load vector are given.Finally,the SASE is used for dynamic fracture analyses of single material planes containing cracks and V-notches.(4)Plane dynamic problem of structure with bi-material interface crack is discussed.Firstly,through the precise algorithm in time domain,governing equations and boundary conditions are transformed into recursive forms.Then the recursive FEM-based formulation is derived through weighted residual method.Secondly,SASE is constructed based on symplectic eigen solutions and special solutions for plane static fracture problem.Element stiffness matrix,Element mass matrix and Element load vector are given.Finally,the SASE is used for dynamic fracture analyses of bi-material interface crack problems.The accuracy and stability of the proposed method in this dissertation is verified by different kinds of numerical simulations,showing that it is an efficient numerical method for fracture dynamics problems.Within each time interval,a precise algorithm in time domain is carried out,in which an adaptive expansion scheme is implemented for guaranteeing the solving accuracy.In space domain,the vicinity of stress singular point is represented by SASE,while the remaining domain is discretized by conventional finite elements.SASE is a kind of displacement-based element,hence it has good universality and compatibility,and can be connected to conventional finite elements directly without any transition elements.Meanwhile,the element is constructed by analytical symplectic eigen solutions,hence fracture parameters,such as stress intensity factor,can be obtained directly without any post-processing.Moreover,the use of SASE can avoid mesh refinement and improve solving efficiency.